Unique solution to an integral equation? (Moment generating function)

Question

I have a solution to the following integral equation:

\begin{equation} \gamma(t) = \dfrac{2}{t}\int_{0}^{1} \gamma(t x)(e^{(1-x)t}-1)dx, \, \gamma(0)=1. \end{equation}

Can one show that the solution is unique? I am only in interested in positive, analytic, increasing solutions. ($\gamma$ is a moment generating function for a [0, 1]- valued random variable.)

I know nothing about integral equations.

Answer 1

Figured out the answer: Make the substitution (suggested by a colleague) $u = tx$. If we differentiate the equation twice, we get a second-order linear differential equation. I know $\gamma'(0) = 1/2$ (the expected value of the random variable), so the solution is indeed unique.